Forces in 2D - Vector Notation (AQA A Level Maths: Mechanics)

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Forces in 2D - Vector Notation

How are forces and vectors linked?

  • Forces are vectors – they have magnitude and direction.
  • The Magnitude of a force is measured in Newtons and the direction of a force is an angle, usually measured in degrees, anti-clockwise from the horizontal
  • There are other ways to talk about direction – in particular Cartesian coordinates(x, y) are used to describe the position of a point in two-dimensional space (plane), relative to a fixed origin O.
  • You may see questions start with a sentence along the lines of “Relative to the frame of reference Oxy …” – this means the two-dimensional space (plane) and distances within it will be based on horizontal (x) and vertical (y) components relative to the origin O.

What is vector notation?

  • Both column vectors and i, j notation can be used for calculating resultant vectors
  • In both cases, a vector (force) is written with its two components separated so the magnitude and direction of the vector are not directly known.

What are the notations for magnitude and direction?

  • The magnitude of the force F N would be denoted by |F| N  or F N  – notice the use of bold and italics in particular
  • Direction is an angle, usually measured in degrees anti-clockwise from the horizontal, θ° is usually used

How do I find the magnitude and direction of a force from its components?

  • For a force F = xi + yj N to find
    • its magnitude,|F| N or F N ,use Pythagoras’ theorem
    • its direction (as an angle), use a diagram and trigonometry

    3-1-4-fig1-finding-the-magnitude-and-direction

  • If either (or both) components are negative then still use a diagram and trigonometry but
    • treat x and y as positive (so |x| and |y| strictly speaking)
    • the angle found may need adjusting depending on where, and which way, the direction is being measured from
    • e.g. Find the direction at which the force F = (-8i - 6j) N, giving your answer as an angle measured in degrees anti-clockwise from the positive horizontal direction

    3-1-4-fig2-negative-components

What does equilibrium with vectors mean?

  • In two dimensions a particle is in equilibrium if the resultant force acting on it in both directions is zero
    • For column vectors, a resultant force of zero would look like open parentheses table row 0 row 0 end table close parentheses
    • For vectors in i-j notation, a resultant force of zero would look like (0i + 0j) N
    • Both forms may be written as 0 N (called ‘the zero vector’)

Worked example

3.1.4_WE_Forces in 2D – Vector Notation_1

   (i)     Find the resultant force.

   (ii)    Find the magnitude of the resultant force and its direction as an angle measured
           anti-clockwise from the i-direction.

   (iii)   A third force is applied to the particle such that it is brought into equilibrium.
           Find the third force, giving your answer in the form left parenthesis x bold i plus y bold j right parenthesis space straight N

3-1-4-fig3-we-solution

Examiner Tip

  • Ideally you should stick to the i-,j- vector notation used in a question but if you prefer one over the other you can use a mixture within the same question.  Just be careful your final answer is in the correct format if requested.
  • Draw diagrams – including ‘mini’-diagrams of individual forces/vectors – this can help in understanding a problem and being accurate.

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.